Erdélyi-Kober Fractional Integrals in the Real Scalar Variable Case
This monograph will examine a new definition for fractional integrals in terms of the distributions of products and ratios of statistically independently distributed positive scalar random variables or in terms of Mellin convolutions of products and ratios in the case of real scalar variables. The idea will be generalized to cover real multivariate cases as well as to real matrix-variate cases. In the matrix-variate case, M-convolutions of products and ratios will be used to extend the ideas. Then we will give a definition for the case of real-valued scalar functions of several real matrices. Then we examine fractional calculus in the complex domain. Here p × p Hermitian positive definite matrices and real-valued scalar functions of these matrices are examined to define and evaluate fractional integrals. It is shown that one can define all types of fractional integrals and fractional derivatives through Erdélyi-Kober fractional integral operators and statistical distribution theory. Then differential operators will be defined by using the following argument. If D−α denotes fractional integral of order α then Dα will be called fractional derivative of order α. For n = 1, 2, …, Dn denotes the integer order derivatives. For \(\Re (n-\alpha )>0\) we can define Dα = DnD−(n−α) or Dα = D−(n−α)Dn. The first will be fractional derivative of order α in the Riemann-Liouville sense and the latter in the Caputo sense. In the present chapter we concentrate on fractional integrals in the real scalar cases.
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